Noncommutative Solitons and Quasideterminants

TL;DR

This paper extends soliton theory to noncommutative spaces, presenting exact solutions using quasideterminants and exploring reductions to integrable equations within noncommutative anti-self-dual Yang-Mills frameworks.

Contribution

It introduces the use of quasideterminants for constructing solutions and discusses reductions of noncommutative Yang-Mills equations to integrable systems.

Findings

Exact solutions via Riemann-Hilbert problem for Atiyah-Ward ansatz

Backlund transformations for noncommutative anti-self-dual Yang-Mills equations

Quasideterminants are crucial in noncommutative solution construction

Abstract

We discuss extension of soliton theory and integrable systems to noncommutative spaces, focusing on integrable aspects of noncommutative anti-self-dual Yang-Mills equations. We give wide class of exact solutions by solving a Riemann-Hilbert problem for the Atiyah-Ward ansatz and present Backlund transformations for the G=U(2) noncommutative anti-self-dual Yang-Mills equations. We find that one kind of noncommutative determinants, quasideterminants, play crucial roles in the construction of noncommutative solutions. We also discuss reduction of a noncommutative anti-self-dual Yang-Mills equation to noncommutative integrable equations. This is partially based on collaboration with C. Gilson and J. Nimmo (Glasgow).